Descriptive Statistics Practice1 [84 marks]

1a.

A group of 20 students travelled to a gymnastics tournament together. Their ages,in years, are given in the following table.

For the students in this group find the mean age;

Markscheme (M1)

Note: Award (M1) for correct substitutions into mean formula.

(A1) (C2)[2 marks]

14+2×15+7×16+17+4×18+19+20+3×2220

(=) 17.5

1b. For the students in this group write down the median age.

Markscheme16.5 (A1) (C1)[1 mark]

[2 marks]

[1mark]

1c.

The lower quartile of the ages is 16 and the upper quartile is 18.5.

Draw a box-and-whisker diagram, for these students’ ages, on thefollowing grid.

Markscheme

(A1)(A1)(A1)(ft) (C3)

Note: Award (A1) for correct endpoints, (A1) for correct quartiles, (A1)(ft)for their median. Follow through from part (a)(ii), but only if median is between16 and 18.5. If a horizontal line goes through the box, award at most (A1)(A1)(A0). Award at most (A0)(A1)(A1) if a ruler has not been used. [3 marks]

[3 marks]

2a.

The lengths of trout in a fisherman’s catch were recorded over one month, and arerepresented in the following histogram.

Complete the following table.

Markscheme

(A2) (C2)

Note: Award (A2) for all correct entries, (A1) for 3 correct entries. [2 marks]

2b. State whether length of trout is a continuous or discrete variable.

[2 marks]

[1mark]

Markschemecontinuous (A1) (C1)[1 mark]

2c. Write down the modal class.

Markscheme (A1) (C1)

Note: Accept equivalent notation such as or .Award (A0) for “60-70” (incorrect notation). [1 mark]

60 (cm) < trout length ⩽ 70 (cm)

(60, 70] ]60, 70]

2d. Any trout with length 40 cm or less is returned to the lake.Calculate the percentage of the fisherman’s catch that is returned to the lake.

Markscheme (M1)

Note: Award (M1) for their 4 divided by their 22.

(A1)(ft) (C2) Note: Follow through from their part (a). Do not accept 0.181818…. [2 marks]

× 100422

= 18.2 (18.1818 …)

[1mark]

[2 marks]

3a.

A tetrahedral (four-sided) die has written on it the numbers 1, 2, 3 and 4. The die isrolled many times and the scores are noted. The table below shows the resultingfrequency distribution.

The die was rolled a total of 100 times.

Write down an equation, in terms of and , for the total number of timesthe die was rolled.

Markscheme or equivalent (A1) (C1)

[1 mark]

x y

18 + x + y + 22 = 100

3b.

The mean score is 2.71.

Using the mean score, write down a second equation in terms of and .

Markscheme or equivalent (M1)(A1) (C2)

Note: Award (M1) for a sum including and , divided by 100 and equatedto 2.71, (A1) for a correct equation. [2 marks]

x y

= 2.7118+2x+3y+88100

x y

3c. Find the value of and of .x y

[1mark]

[2 marks]

[3 marks]

Markscheme and (M1)

Note: Award (M1) for obtaining a correct linear equation in one variablefrom their (a) and their (b).This may be implied if seen in part (a) or part (b).

(A1)(ft)(A1)(ft) (C3) Notes: Follow through from parts (a) and (b), irrespective of working seenprovided the answers are positive integers. [3 marks]

x + y = 60 2x + 3y = 165

x = 15; y = 45

4a. A group of students were asked how long they spend practisingmathematics during the week. The results are shown in the followingtable.

It is known that .Write downi) the modal class;ii) the mid-interval value of the modal class;iii) the class in which the median lies.

35 < a < 52

[3 marks]

Markschemei) (A1) (C1)Note: Accept equivalent notation: or . ii) (A1)(ft) (C1)Note: Follow through from part (a)(i). iii) (A1)(ft) (C1)Note: Follow through from part (a)(i), for consistent misuse of inequality.Accept equivalent notation: or .

3 ⩽ t < 4

[3, 4) [3, 4[

3.5

2 ⩽ t < 3

[2, 3) [2, 3[

4b. For this group of students, the estimated mean number of hours spentpractising mathematics is .Calculate the value of .

Markscheme (M1)(A1)(ft)

Notes: Award (M1) for substitution into mean formula and equating to ,(A1)(ft) for correct substitutions. Follow through from their mid-interval valuein part (a)(ii).

(A1)(ft) (C3)Note: The final (A1)(ft) is awarded only if is an integer and .Follow through from part (a)(ii).

2.69a

= 2.693.5×0.5+30×1.5+a×2.5+52×3.5+43×4.535+30+a+52+43

2.69

(a =) 40

a 35 < a < 52

5a.

The IB grades attained by a group of students are listed as follows.

Find the median grade.

6 4 5 3 7 3 5 4 2 5

[3 marks]

[2 marks]

Markscheme (M1)

Note: Award (M1) for correct ordered set.

(A1) (C2)

2 3 3 4 4 5 5 5 6 7

(Median =) 4.5

5b. Calculate the interquartile range.

Markscheme (M1)

Note: Award (M1) for correct quartiles seen.

(A1) (C2)

5 − 3

= 2

5c. Find the probability that a student chosen at random from the groupscored at least a grade .

Markscheme (A2) (C2)

4

(0.7, 70%)710

[2 marks]

[2 marks]

6a.

Two groups of 40 students were asked how many books they have read in the lasttwo months. The results for the first group are shown in the following table.

The quartiles for these results are 3 and 5.

Write down the value of the median for these results.

Markscheme (A1)(C1)4

6b. Draw a box-and-whisker diagram for these results on the following grid.

[1mark]

[3 marks]

Markscheme

(A1)(ft)(A1)

(A1) (C3) Notes: Award (A1)(ft) for correct median, (A1) for correct quartiles and box,(A1) for endpoints 2 and 8 joined by a straight line that does not cross the box.Follow through from their median from part (a).

6c. The results for the second group of 40 students are shown in thefollowing box-and-whisker diagram.

Estimate the number of students in the second group who have read at least 6books.

Markscheme (M1)

Notes: Award (M1) for OR .

(A1) (C2)

40 × 0.25

40 × 25% 40 − 40 × 75%

10

[2 marks]

7a.

The time, in minutes, that students in a school spend on their homework per day ispresented in the following box-and-whisker diagram.

Time, in minutes, students spend on their homework per day

Find(i) the longest amount of time spent on homework per day;(ii) the interquartile range.

Markscheme(i) 300 (minutes) OR 5 hours (A1) Note: If answer given in hours, the unit must be seen.(ii) 220 – 100 (M1)Notes: Award (M1) for the two quartiles seen.= 120 (minutes) OR 2 hours (A1) (C3)Note: If answer given in hours, the unit must be seen.

7b. State the statistical term corresponding to the value of 140 minutes.

Markschememedian (time spent on homework per day) (A1) (C1)Note: Do not accept middle or medium etc.

7c. Find the percentage of students who spend(i) between 100 and 140 minutes per day on their homework;(ii) more than 100 minutes per day on their homework.

[3 marks]

[1mark]

[2 marks]

Markscheme(i) 25 (A1)(ii) 75 (A1) (C2)

8a.

In a particular week, the number of eggs laid by each hen on a farm was counted.The results are summarized in the following table.

State whether these data are discrete or continuous.

Markschemediscrete (A1) (C1)

8b. Write down(i) the number of hens on the farm;(ii) the modal number of eggs laid.

Markscheme(i) 60 (A1)(ii) 5 (A1) (C2)

8c. Calculate(i) the mean number of eggs laid;(ii) the standard deviation.

[1mark]

[2 marks]

[3 marks]

Markscheme(i) (M1)

Notes: Award (M1) for an attempt to substitute into the “mean of a set ofdata” formula, with at least three correct terms in the numerator.Denominator must be 60.Follow through from part (b)(i), only if work is seen.

(A1)Notes: Award at most (M1)(A0) for an answer of 4 but only if working seen. (ii) (A1) (C3)

1×4+2×7+3×12…60

= 4.03 (4.03333 …)

1.54 (1.53803 …)

9a.

A class of 13 Mathematics students received the following grades in their final IB examination.

3 5 3 4 7 3 2 7 5 6 5 3 4

For these grades, find the mode;

Markscheme3 (A1) (C1)[1 mark]

9b. For these grades, find the median;

Markscheme4 (M1)(A1) (C2) Note: Award (M1) for ordered list of numbers seen. [2 marks]

[1mark]

[2 marks]

9c. For these grades, find the upper quartile;

Markscheme5.5 (A1) (C1)[1 mark]

9d. For these grades, find the interquartile range.

Markscheme5.5 – 3 (M1) Note: Award (M1) for 3 and their 5.5 seen. = 2.5 (A1)(ft) (C2) Note: Follow through from their answer to part (c). [2 marks]

10a.

The distribution of rainfall in a town over 80 days is displayed on the following box-and-whisker diagram.

Write down the median rainfall.

[1mark]

[2 marks]

[1mark]

Markscheme (A1) (C1)

[1 mark]43 (mm)

10b. Write down the minimum rainfall.

Markscheme (A1) (C1)

[1 mark]10 (mm)

10c. Find the interquartile range.

Markscheme (A1)

(A1) (C2) Note: Award (A1) for identifying correct quartiles, (A1) for correct subtractionof the quartiles. [2 marks]

48 − 20= 28

10d. Write down the number of days the rainfall will be(i) between 43 mm and 48 mm;(ii) between 20 mm and 59 mm.

[1mark]

[2 marks]

[2 marks]

Markscheme(i) 20 (days) (A1)(ii) 60 (days) (A1) (C2)[2 marks]

11a.

A survey was carried out on a road to determine the number of passengers in each car (excluding thedriver). The table shows the results of the survey.

State whether the data is discrete or continuous.

Markschemediscrete (A1) (C1)[1 mark]

11b. Write down the mode.

Markscheme (A1) (C1)

[1 mark]0

11c. Use your graphic display calculator to find(i) the mean number of passengers per car;(ii) the median number of passengers per car;(iii) the standard deviation.

[1mark]

[1mark]

[4 marks]

Markscheme(i) (A2) Note: Award (M1) for seen. Accept or as a final answer if or seen. (ii) (A1)(iii) (A1) (C4)[4 marks]

1.47 (1.46666...)

176120

1 2 1.4666 … 1.47

1.5

1.25 (1.25122...)

12a.

The weights, in kg, of 60 adolescent females were collected and are summarized in the box and whiskerdiagram shown below.

Write down the median weight of the females.

Markscheme42 kg (A1) (C1)

Note: The units are required.

12b. Calculate the range.

[1mark]

[2 marks]

Markscheme58 − 33 (A1)

Note: Award (A1) for correct maximum and minimum seen.

= 25 (A1) (C2)

12c. Estimate the probability that the weight of a randomly chosen female ismore than 50 kg.

Markscheme (A1) (C1)(0.25, 25%)1

4

12d. Use the box and whisker diagram to determine if the mean weight of thefemales is less than the median weight. Give a reason for your answer.

MarkschemeMean weight is more than the median weight. (A1)The upper half of the distribution is wider (more dispersed) or data is positively(or right) skewed or equivalent reason. (R1)OR

(R1) (C2)

Note: Do not award (A1)(R0).

(The mean is calculated x̄ = )35.5×15+40×15+54×1560

x̄ = 43.875 (kg)

[1mark]

[2 marks]

13a.

The number of passengers in the first ten carriages of a train is listed below.

6 , 8 , 6 , 3 , 8 , 4 , 8 , 5 , p , p

The mean number of passengers per carriage is 5.6.

Calculate the value of p.

Markscheme (M1)

Notes: Accept equivalent forms. Award (M1) for correct substitutions in meanformula.

4 (A1) (C2)

= 5.648+2p

10

13b. Find the median number of passengers per carriage.

MarkschemeCorrectly rearranging the list with their p (M1)5.5 (A1)(ft) (C2)

Note: Follow through from their value of p in part (a).

13c. If the passengers in the eleventh carriage are also included, the meannumber of passengers per carriage increases to 6.0.

Determine the number of passengers in the eleventh carriage of the train.

[2 marks]

[2 marks]

[2 marks]

Markscheme (M1)

Notes: Accept equivalent forms. Award (M1) for correct substitutions in meanformula.

OR

(M1)10 (A1)(ft) (C2)

Note: Follow through from their answer to part (a).

= 6.056+x11

48+2× their part (a)+x

11

14a.

Toronto’s annual snowfall, x, in cm, has been recorded for the past 176 years. The results are shown inthe table.

Write down the modal class.

Markscheme14 ≤ x < 18 (A1) (C1)[1 mark]

14b. Write down the mid interval value for the class 6 ≤ x < 10 .

Markscheme8 (A1) (C1)[1 mark]

[1mark]

[1mark]

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14c. Calculate an estimate of the mean annual snowfall.

Markscheme (M1)

Notes: Award (M1) for an attempt to substitute their mid-interval values(consistent with their answer to part (b)) into the formula for the mean. Award(M1) where a table is constructed with their (consistent) mid-interval valueslisted along with the frequencies. = 14.7 (cm) (14.7045…) (A1)(ft) (C2)Notes: Follow through from their answer to part (b). If a final incorrect answerthat is consistent with their (b) is given award (M1)(A1)(ft) even if no workingis seen.[2 marks]

4×30 + 8×26 + 12×29 + 16×32 + 20×18 + 24×27 + 28×14176

14d. Find the number of years for which the annual snowfall was at least 18cm.

Markscheme18 + 27 + 14 (M1)Note: Award (M1) for adding 18, 27 and 14. = 59 (A1) (C2)[2 marks]

[2 marks]

[2 marks]